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#### Flashcard 1731707669772

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#finance
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the Black–Scholes formula estimates the price of [...]

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is a mathematical model of a financial market containing derivative investment instruments. From the partial differential equation in the model, known as the Black–Scholes equation, one can deduce the Black–Scholes formula, which gives <span>a theoretical estimate of the price of European-style options and shows that the option has a unique price regardless of the risk of the security and its expected return (instead replacing the security's expected return with the risk-neutral rate)

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Black–Scholes model - Wikipedia
Black–Scholes model - Wikipedia Black–Scholes model From Wikipedia, the free encyclopedia (Redirected from Black–Scholes) Jump to: navigation, search The Black–Scholes /ˌblæk ˈʃoʊlz/ [1] or Black–Scholes–Merton model is a mathematical model of a financial market containing derivative investment instruments. From the partial differential equation in the model, known as the Black–Scholes equation, one can deduce the Black–Scholes formula, which gives a theoretical estimate of the price of European-style options and shows that the option has a unique price regardless of the risk of the security and its expected return (instead replacing the security's expected return with the risk-neutral rate). The formula led to a boom in options trading and provided mathematical legitimacy to the activities of the Chicago Board Options Exchange and other options markets around the world. [2]

#### Annotation 1732632251660

 #mathematical-structures In many fields of mathematics, morphism refers to a structure-preserving map from one mathematical structure to another. The notion of morphism recurs in much of contemporary mathematics. In set theory, morphisms are functions; in linear algebra, linear transformations; in group theory, group homomorphisms; in topology, continuous functions, and so on.

Morphism - Wikipedia
st of references, but its sources remain unclear because it has insufficient inline citations. Please help to improve this article by introducing more precise citations. (April 2016) (Learn how and when to remove this template message) <span>In many fields of mathematics, morphism refers to a structure-preserving map from one mathematical structure to another. The notion of morphism recurs in much of contemporary mathematics. In set theory, morphisms are functions; in linear algebra, linear transformations; in group theory, group homomorphisms; in topology, continuous functions, and so on. In category theory, morphism is a broadly similar idea, but somewhat more abstract: the mathematical objects involved need not be sets, and the relationship between them may be someth

#### Annotation 1732634348812

 #mathematical-structures In many fields of mathematics, morphism refers to a structure-preserving map from one mathematical structure to another.

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In many fields of mathematics, morphism refers to a structure-preserving map from one mathematical structure to another. The notion of morphism recurs in much of contemporary mathematics. In set theory, morphisms are functions; in linear algebra, linear transformations; in group theory, group homomorphism

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Morphism - Wikipedia
st of references, but its sources remain unclear because it has insufficient inline citations. Please help to improve this article by introducing more precise citations. (April 2016) (Learn how and when to remove this template message) <span>In many fields of mathematics, morphism refers to a structure-preserving map from one mathematical structure to another. The notion of morphism recurs in much of contemporary mathematics. In set theory, morphisms are functions; in linear algebra, linear transformations; in group theory, group homomorphisms; in topology, continuous functions, and so on. In category theory, morphism is a broadly similar idea, but somewhat more abstract: the mathematical objects involved need not be sets, and the relationship between them may be someth

#### Flashcard 1732635921676

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#mathematical-structures
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[...] refers to a structure-preserving map from one mathematical structure to another.
morphism

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In many fields of mathematics, morphism refers to a structure-preserving map from one mathematical structure to another.

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Morphism - Wikipedia
st of references, but its sources remain unclear because it has insufficient inline citations. Please help to improve this article by introducing more precise citations. (April 2016) (Learn how and when to remove this template message) <span>In many fields of mathematics, morphism refers to a structure-preserving map from one mathematical structure to another. The notion of morphism recurs in much of contemporary mathematics. In set theory, morphisms are functions; in linear algebra, linear transformations; in group theory, group homomorphisms; in topology, continuous functions, and so on. In category theory, morphism is a broadly similar idea, but somewhat more abstract: the mathematical objects involved need not be sets, and the relationship between them may be someth

#### Flashcard 1732637494540

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#mathematical-structures
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morphism refers to a [...] from one mathematical structure to another.
structure-preserving map

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In many fields of mathematics, morphism refers to a structure-preserving map from one mathematical structure to another.

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Morphism - Wikipedia
st of references, but its sources remain unclear because it has insufficient inline citations. Please help to improve this article by introducing more precise citations. (April 2016) (Learn how and when to remove this template message) <span>In many fields of mathematics, morphism refers to a structure-preserving map from one mathematical structure to another. The notion of morphism recurs in much of contemporary mathematics. In set theory, morphisms are functions; in linear algebra, linear transformations; in group theory, group homomorphisms; in topology, continuous functions, and so on. In category theory, morphism is a broadly similar idea, but somewhat more abstract: the mathematical objects involved need not be sets, and the relationship between them may be someth

#### Annotation 1735703268620

 #topology In mathematics, topology (from the Greek τόπος, place, and λόγος, study) is concerned with the properties of space that are preserved under continuous deformations, such as stretching, crumpling and bending, but not tearing or gluing.

Topology - Wikipedia
ogy (disambiguation). For a topology of a topos or category, see Lawvere–Tierney topology and Grothendieck topology. [imagelink] Möbius strips, which have only one surface and one edge, are a kind of object studied in topology. <span>In mathematics, topology (from the Greek τόπος, place, and λόγος, study) is concerned with the properties of space that are preserved under continuous deformations, such as stretching, crumpling and bending, but not tearing or gluing. This can be studied by considering a collection of subsets, called open sets, that satisfy certain properties, turning the given set into what is known as a topological space. Important

#### Flashcard 1741389171980

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#topology
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topology concerns the properties of space that are preserved under [...]
continuous deformations

i.e. homeomorphisms. Remeber scrub

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In mathematics, topology (from the Greek τόπος, place, and λόγος, study) is concerned with the properties of space that are preserved under continuous deformations, such as stretching, crumpling and bending, but not tearing or gluing.

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Topology - Wikipedia
ogy (disambiguation). For a topology of a topos or category, see Lawvere–Tierney topology and Grothendieck topology. [imagelink] Möbius strips, which have only one surface and one edge, are a kind of object studied in topology. <span>In mathematics, topology (from the Greek τόπος, place, and λόγος, study) is concerned with the properties of space that are preserved under continuous deformations, such as stretching, crumpling and bending, but not tearing or gluing. This can be studied by considering a collection of subsets, called open sets, that satisfy certain properties, turning the given set into what is known as a topological space. Important

#### Flashcard 1741390744844

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#topology
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topology is concerned with [...] that are preserved under continuous deformations
the properties of space

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In mathematics, topology (from the Greek τόπος, place, and λόγος, study) is concerned with the properties of space that are preserved under continuous deformations, such as stretching, crumpling and bending, but not tearing or gluing.

#### Original toplevel document

Topology - Wikipedia
ogy (disambiguation). For a topology of a topos or category, see Lawvere–Tierney topology and Grothendieck topology. [imagelink] Möbius strips, which have only one surface and one edge, are a kind of object studied in topology. <span>In mathematics, topology (from the Greek τόπος, place, and λόγος, study) is concerned with the properties of space that are preserved under continuous deformations, such as stretching, crumpling and bending, but not tearing or gluing. This can be studied by considering a collection of subsets, called open sets, that satisfy certain properties, turning the given set into what is known as a topological space. Important

#### Annotation 1744134606092

 #vector-space A norm must also satisfy certain properties pertaining to scalability and additivity which are given in the formal definition below.

Norm (mathematics) - Wikipedia
ositive length or size to each vector in a vector space—save for the zero vector, which is assigned a length of zero. A seminorm, on the other hand, is allowed to assign zero length to some non-zero vectors (in addition to the zero vector). <span>A norm must also satisfy certain properties pertaining to scalability and additivity which are given in the formal definition below. A simple example is two dimensional Euclidean space R 2 equipped with the "Euclidean norm" (see below) Elements in this vector space (e.g., (3, 7)) are usually drawn as arr

#### Flashcard 1744136703244

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#vector-space
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A norm must also satisfy certain properties pertaining to [...property...]

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A norm must also satisfy certain properties pertaining to scalability and additivity which are given in the formal definition below.

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Norm (mathematics) - Wikipedia
ositive length or size to each vector in a vector space—save for the zero vector, which is assigned a length of zero. A seminorm, on the other hand, is allowed to assign zero length to some non-zero vectors (in addition to the zero vector). <span>A norm must also satisfy certain properties pertaining to scalability and additivity which are given in the formal definition below. A simple example is two dimensional Euclidean space R 2 equipped with the "Euclidean norm" (see below) Elements in this vector space (e.g., (3, 7)) are usually drawn as arr

#### Annotation 1749118225676

 #matrix-decomposition By the spectral theorem, real symmetric matrices and complex Hermitian matrices have an eigenbasis; that is, every vector is expressible as a linear combination of eigenvectors. In both cases, all eigenvalues are real.

Matrix (mathematics) - Wikipedia
x. In complex matrices, symmetry is often replaced by the concept of Hermitian matrices, which satisfy A ∗ = A, where the star or asterisk denotes the conjugate transpose of the matrix, that is, the transpose of the complex conjugate of A. <span>By the spectral theorem, real symmetric matrices and complex Hermitian matrices have an eigenbasis; that is, every vector is expressible as a linear combination of eigenvectors. In both cases, all eigenvalues are real. [29] This theorem can be generalized to infinite-dimensional situations related to matrices with infinitely many rows and columns, see below. Invertible matrix and its inverse[edit s

#### Annotation 1749124779276

 #matrix-decomposition By the spectral theorem, real symmetric matrices and complex Hermitian matrices have an eigenbasis

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By the spectral theorem, real symmetric matrices and complex Hermitian matrices have an eigenbasis; that is, every vector is expressible as a linear combination of eigenvectors. In both cases, all eigenvalues are real.

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Matrix (mathematics) - Wikipedia
x. In complex matrices, symmetry is often replaced by the concept of Hermitian matrices, which satisfy A ∗ = A, where the star or asterisk denotes the conjugate transpose of the matrix, that is, the transpose of the complex conjugate of A. <span>By the spectral theorem, real symmetric matrices and complex Hermitian matrices have an eigenbasis; that is, every vector is expressible as a linear combination of eigenvectors. In both cases, all eigenvalues are real. [29] This theorem can be generalized to infinite-dimensional situations related to matrices with infinitely many rows and columns, see below. Invertible matrix and its inverse[edit s

#### Flashcard 1749126352140

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#matrix-decomposition
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By the spectral theorem, real symmetric matrices and complex Hermitian matrices have an [...]

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By the spectral theorem, real symmetric matrices and complex Hermitian matrices have an eigenbasis

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Matrix (mathematics) - Wikipedia
x. In complex matrices, symmetry is often replaced by the concept of Hermitian matrices, which satisfy A ∗ = A, where the star or asterisk denotes the conjugate transpose of the matrix, that is, the transpose of the complex conjugate of A. <span>By the spectral theorem, real symmetric matrices and complex Hermitian matrices have an eigenbasis; that is, every vector is expressible as a linear combination of eigenvectors. In both cases, all eigenvalues are real. [29] This theorem can be generalized to infinite-dimensional situations related to matrices with infinitely many rows and columns, see below. Invertible matrix and its inverse[edit s

#### Annotation 1753204526348

 #vector-space a norm is a function that assigns a strictly positive length or size to each vector (bar zero vector) in a vector space

Norm (mathematics) - Wikipedia
analysis. For field theory, see Field norm. For ideals, see Ideal norm. For group theory, see Norm (group). For norms in descriptive set theory, see prewellordering. In linear algebra, functional analysis, and related areas of mathematics, <span>a norm is a function that assigns a strictly positive length or size to each vector in a vector space—save for the zero vector, which is assigned a length of zero. A seminorm, on the other hand, is allowed to assign zero length to some non-zero vectors (in addition to the zero vector).

#### Flashcard 1753207409932

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#vector-space
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a norm is a [...] that assigns a strictly positive length or size to each vector in a vector space
function

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a norm is a function that assigns a strictly positive length or size to each vector in a vector space

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Norm (mathematics) - Wikipedia
analysis. For field theory, see Field norm. For ideals, see Ideal norm. For group theory, see Norm (group). For norms in descriptive set theory, see prewellordering. In linear algebra, functional analysis, and related areas of mathematics, <span>a norm is a function that assigns a strictly positive length or size to each vector in a vector space—save for the zero vector, which is assigned a length of zero. A seminorm, on the other hand, is allowed to assign zero length to some non-zero vectors (in addition to the zero vector).

#### Flashcard 1753209769228

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#vector-space
Question
a norm is a function that assigns a [...] to each vector in a vector space
nonnegative length or size

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a norm is a function that assigns a strictly positive length or size to each vector (bar zero vector) in a vector space

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Norm (mathematics) - Wikipedia
analysis. For field theory, see Field norm. For ideals, see Ideal norm. For group theory, see Norm (group). For norms in descriptive set theory, see prewellordering. In linear algebra, functional analysis, and related areas of mathematics, <span>a norm is a function that assigns a strictly positive length or size to each vector in a vector space—save for the zero vector, which is assigned a length of zero. A seminorm, on the other hand, is allowed to assign zero length to some non-zero vectors (in addition to the zero vector).

#### Annotation 1758183427340

 #differential-equations In mathematics, a linear differential equation is a differential equation that is defined by a linear polynomial in the unknown function and its derivatives

Linear differential equation - Wikipedia
tml>Linear differential equation - Wikipedia Linear differential equation From Wikipedia, the free encyclopedia Jump to: navigation, search In mathematics, a linear differential equation is a differential equation that is defined by a linear polynomial in the unknown function and its derivatives, that is an equation of the form a 0 ( x ) y +

#### Flashcard 1758185524492

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#differential-equations
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a linear differential equation is defined by a [...] in the unknown function and its derivatives

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In mathematics, a linear differential equation is a differential equation that is defined by a linear polynomial in the unknown function and its derivatives

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Linear differential equation - Wikipedia
tml>Linear differential equation - Wikipedia Linear differential equation From Wikipedia, the free encyclopedia Jump to: navigation, search In mathematics, a linear differential equation is a differential equation that is defined by a linear polynomial in the unknown function and its derivatives, that is an equation of the form a 0 ( x ) y +

#### Flashcard 1758187097356

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#differential-equations
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a [...] is a differential equation defined by a linear polynomial in the unknown function and its derivatives
linear differential equation

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In mathematics, a linear differential equation is a differential equation that is defined by a linear polynomial in the unknown function and its derivatives

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Linear differential equation - Wikipedia
tml>Linear differential equation - Wikipedia Linear differential equation From Wikipedia, the free encyclopedia Jump to: navigation, search In mathematics, a linear differential equation is a differential equation that is defined by a linear polynomial in the unknown function and its derivatives, that is an equation of the form a 0 ( x ) y +

#### Flashcard 1758518185228

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#topology
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continuous deformations includes [...] , but not tearing or gluing.
streching, crumpling and bending

Mind the scrub!

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In mathematics, topology (from the Greek τόπος, place, and λόγος, study) is concerned with the properties of space that are preserved under continuous deformations, such as stretching, crumpling and bending, but not tearing or gluing.

#### Original toplevel document

Topology - Wikipedia
ogy (disambiguation). For a topology of a topos or category, see Lawvere–Tierney topology and Grothendieck topology. [imagelink] Möbius strips, which have only one surface and one edge, are a kind of object studied in topology. <span>In mathematics, topology (from the Greek τόπος, place, and λόγος, study) is concerned with the properties of space that are preserved under continuous deformations, such as stretching, crumpling and bending, but not tearing or gluing. This can be studied by considering a collection of subsets, called open sets, that satisfy certain properties, turning the given set into what is known as a topological space. Important

#### Annotation 1758527884556

 #measure-theory Any countable set of real numbers has Lebesgue measure 0.

Lebesgue measure - Wikipedia
c, d] is Lebesgue measurable, and its Lebesgue measure is (b − a)(d − c), the area of the corresponding rectangle. Moreover, every Borel set is Lebesgue measurable. However, there are Lebesgue measurable sets which are not Borel sets. [3] [4] <span>Any countable set of real numbers has Lebesgue measure 0. In particular, the Lebesgue measure of the set of rational numbers is 0, although the set is dense in R. The Cantor set is an example of an uncountable set that has Lebesgue measure zer

#### Annotation 1758529981708

 #measure-theory The Lebesgue outer measure of a set E emerges as the greatest lower bound (infimum) of the lengths from among all possible such sets (unions of open intervals that include E).

Lebesgue measure - Wikipedia
, because E {\displaystyle E} is a subset of the union of the intervals, and so the intervals may include points which are not in E {\displaystyle E} . <span>The Lebesgue outer measure emerges as the greatest lower bound (infimum) of the lengths from among all possible such sets. Intuitively, it is the total length of those interval sets which fit E {\displaystyle E} most tightly and do not overlap. That characterize

#### Flashcard 1758533389580

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#measure-theory
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The Lebesgue outer measure of a set E emerges as [...] of the lengths from among all possible such sets (unions of open intervals that include E).

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The Lebesgue outer measure of a set E emerges as the greatest lower bound (infimum) of the lengths from among all possible such sets (unions of open intervals that include E).

#### Original toplevel document

Lebesgue measure - Wikipedia
, because E {\displaystyle E} is a subset of the union of the intervals, and so the intervals may include points which are not in E {\displaystyle E} . <span>The Lebesgue outer measure emerges as the greatest lower bound (infimum) of the lengths from among all possible such sets. Intuitively, it is the total length of those interval sets which fit E {\displaystyle E} most tightly and do not overlap. That characterize

#### Flashcard 1758534962444

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#measure-theory
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The Lebesgue outer measure of a set E is the greatest lower bound of the lengths from among all possible such sets that [...]
unions of open intervals that include E

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The Lebesgue outer measure of a set E emerges as the greatest lower bound (infimum) of the lengths from among all possible such sets (unions of open intervals that include E).

#### Original toplevel document

Lebesgue measure - Wikipedia
, because E {\displaystyle E} is a subset of the union of the intervals, and so the intervals may include points which are not in E {\displaystyle E} . <span>The Lebesgue outer measure emerges as the greatest lower bound (infimum) of the lengths from among all possible such sets. Intuitively, it is the total length of those interval sets which fit E {\displaystyle E} most tightly and do not overlap. That characterize

#### Flashcard 1758536535308

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#measure-theory
Question
[...] of a set E emerges as the greatest lower bound (infimum) of the lengths from among all possible such sets (unions of open intervals that include E).
The Lebesgue outer measure

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The Lebesgue outer measure of a set E emerges as the greatest lower bound (infimum) of the lengths from among all possible such sets (unions of open intervals that include E).

#### Original toplevel document

Lebesgue measure - Wikipedia
, because E {\displaystyle E} is a subset of the union of the intervals, and so the intervals may include points which are not in E {\displaystyle E} . <span>The Lebesgue outer measure emerges as the greatest lower bound (infimum) of the lengths from among all possible such sets. Intuitively, it is the total length of those interval sets which fit E {\displaystyle E} most tightly and do not overlap. That characterize

#### Flashcard 1758538108172

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#measure-theory
Question
Any [...] set of real numbers has Lebesgue measure 0.

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Any countable set of real numbers has Lebesgue measure 0.

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Lebesgue measure - Wikipedia
c, d] is Lebesgue measurable, and its Lebesgue measure is (b − a)(d − c), the area of the corresponding rectangle. Moreover, every Borel set is Lebesgue measurable. However, there are Lebesgue measurable sets which are not Borel sets. [3] [4] <span>Any countable set of real numbers has Lebesgue measure 0. In particular, the Lebesgue measure of the set of rational numbers is 0, although the set is dense in R. The Cantor set is an example of an uncountable set that has Lebesgue measure zer

#### Flashcard 1758539681036

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#measure-theory
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As a set the rational number is [...but...] .
infinite but countable

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#### Annotation 1759647501580

 #vector-space Let p ≥ 1 be a real number. The -norm of vectors is

Norm (mathematics) - Wikipedia
x i {\displaystyle \sum _{i=1}^{n}x_{i}} is not a norm because it may yield negative results. p-norm[edit source] Main article: L p space <span>Let p ≥ 1 be a real number. The p {\displaystyle p} -norm (also called ℓ p {\displaystyle \ell _{p}} -norm) of vectors x = ( x 1 , … , x n ) {\displaystyle \mathbf {x} =(x_{1},\ldots ,x_{n})} is ‖ x ‖ p := ( ∑ i = 1 n | x i | p ) 1 / p . {\displaystyle \left\|\mathbf {x} \right\|_{p}:={\bigg (}\sum _{i=1}^{n}\left|x_{i}\right|^{p}{\bigg )}^{1/p}.} For p = 1 we get the taxicab norm, for p = 2 we get the Euclidean norm, and as p approaches ∞ {\displaystyle \infty } the p-norm approa

#### Flashcard 1759650385164

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#vector-space
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Let p ≥ 1 be a real number. The -norm of vectors is [...]

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Let p ≥ 1 be a real number. The -norm of vectors is

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Norm (mathematics) - Wikipedia
x i {\displaystyle \sum _{i=1}^{n}x_{i}} is not a norm because it may yield negative results. p-norm[edit source] Main article: L p space <span>Let p ≥ 1 be a real number. The p {\displaystyle p} -norm (also called ℓ p {\displaystyle \ell _{p}} -norm) of vectors x = ( x 1 , … , x n ) {\displaystyle \mathbf {x} =(x_{1},\ldots ,x_{n})} is ‖ x ‖ p := ( ∑ i = 1 n | x i | p ) 1 / p . {\displaystyle \left\|\mathbf {x} \right\|_{p}:={\bigg (}\sum _{i=1}^{n}\left|x_{i}\right|^{p}{\bigg )}^{1/p}.} For p = 1 we get the taxicab norm, for p = 2 we get the Euclidean norm, and as p approaches ∞ {\displaystyle \infty } the p-norm approa

#### Flashcard 1759652744460

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#vector-space
Question

Let p ≥ 1 be a real number. The [...] of vectors is